The integer for which is a finite non-zero number, is
- A.1
- B.2
- C.3
- D.4
Show answer and solution
Correct option: C
Solution
Wait - let me recheck. The official answer says option 3 (n=3), but let me verify.
Expand numerator using Taylor series:
- cos x - 1 ≈ -x²/2 + x⁴/24 - ...
- cos x - eˣ ≈ (1 - x²/2 + x⁴/24...) - (1 + x + x²/2 + x³/6...) = -x - x² - x³/6 + ...
Product: (-x²/2 + ...)(-x - x² - x³/6 + ...) = x³/2 + x⁴/2 + ...
So the leading term is x³/2, meaning the limit equals (1/2)/x^(n-3) as x→0. For a finite non-zero limit, we need n = 3, giving the limit = 1/2.
The answer is n = 3 (option 3), because the numerator's leading power is x³, so dividing by x³ gives the finite non-zero value 1/2.
Expand numerator using Taylor series:
- cos x - 1 ≈ -x²/2 + x⁴/24 - ...
- cos x - eˣ ≈ (1 - x²/2 + x⁴/24...) - (1 + x + x²/2 + x³/6...) = -x - x² - x³/6 + ...
Product: (-x²/2 + ...)(-x - x² - x³/6 + ...) = x³/2 + x⁴/2 + ...
So the leading term is x³/2, meaning the limit equals (1/2)/x^(n-3) as x→0. For a finite non-zero limit, we need n = 3, giving the limit = 1/2.
The answer is n = 3 (option 3), because the numerator's leading power is x³, so dividing by x³ gives the finite non-zero value 1/2.
[AIEEE 2002]